Finite Group Databasebeta

← #11781586

Subgroups — #11781586

Order 1152.

320 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 320 conjugacy classes of subgroups — too many to draw. Only its 88 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#5093C211yesmaximal, characteristic
2C3×#5091C211yesmaximal, characteristic
3#10590285C211yesmaximal, characteristic
4C3⋊C8×Dic6C211yesmaximal, characteristic
5#10589412C211yesmaximal, characteristic
6#10589099C211yesmaximal, characteristic
7#10589092C211yesmaximal, characteristic
8#71241—31nomaximal
9#71151—31nomaximal
10C32×C4⋊Q8C2211yescharacteristic
11C3×#869C2211yescharacteristic
12C3×C4×Dic6C2211yescharacteristic
13C4⋊(C3⋊(C3⋊C8))C2211yescharacteristic
14C4⋊(C3⋊(C3⋊C8))C2211yescharacteristic
15C3×#810—21no—
16C3×#810—21no—
17C3×#810—21no—
18C3×#807—21no—
19C3×C4⋊(C3⋊C8)C2211yescharacteristic
20C3×C4×C3⋊C8C2211yescharacteristic
21#63574—21no—
22Dic3⋊(C3⋊C8)—21no—
23#63565—21no—
24Dic3×C3⋊C8—21no—
25C3×#487—31no—
26C3×#477—31no—
27#5094—31no—
28#5093D311yescharacteristic
29#5091D311yescharacteristic
30Q8×C3⋊C8—31no—
31#5020—31no—
32C8×Dic6—31no—
33#4547—31no—
34#4546—31no—
35#4546—31no—
36#4546—61no—
37#4486—31no—
38#4479—31no—
39#4458—31no—
40C2×C32×Q8—21no—
41C2×C3×Dic6—21no—
42C2×C3×Dic6D411yescharacteristic
43C32×C4⋊C4—21no—
44C32×C4⋊C4—21no—
45C32×C42C2311yescharacteristic
46C2×C3⋊(C3⋊C8)—21no—
47C2×C3⋊(C3⋊C8)—21no—
48C3×C4⋊Dic3—21no—
49C3×C4⋊Dic3—21no—
50C3×C4⋊Dic3D411yescharacteristic
51C3×Dic3⋊C4—21no—
52C3×C4×Dic3—21no—
53C2×C3×C3⋊C8—21no—
54C2×C3×C3⋊C8D411yescharacteristic
55C2×C3×C3⋊C8D411yescharacteristic
56#1684—91noSylow
57C3×C4⋊Q8—21no—
58C3×C4⋊Q8—31no—
59C3×C4⋊Q8C2×D311yescharacteristic
60C3×C4⋊Q8C2×D311yescharacteristic
61C3×C4×Q8—31no—
62#869C2×D311yescharacteristic
63C4×Dic6C2×D311yescharacteristic
64C3×#107—61no—
65C3×C4⋊C8—31no—
66C3×#103—61no—
67C3×#103—61no—
68C3×#103—61no—
69C3×C4×C8—31no—
70#835—61no—
71Dic3⋊C8—61no—
72C8×Dic3—61no—
73#810—21no—
74#810—21no—
75#810—21no—
76#810—61no—
77#808—61no—
78#808—61no—
79#807—21no—
80C4⋊(C3⋊C8)—31no—
81C4⋊(C3⋊C8)—31no—
82C4⋊(C3⋊C8)—31no—
83C4⋊(C3⋊C8)—31no—
84C4⋊(C3⋊C8)—61no—
85C4⋊(C3⋊C8)—61no—
86C4⋊(C3⋊C8)—61no—
87C4⋊(C3⋊C8)C2×D311yescharacteristic
88C4×C3⋊C8—61no—
89C4×C3⋊C8C2×D311yescharacteristic
90C32×Q8—81no—
91C2×C32×C4—41no—
92C2×C32×C4C2×D411yescharacteristic
93C2×C32×C4C2×D411yescharacteristic
94C2×C32×C4#4111yescharacteristic
95C2×C3×Dic3—21no—
96C2×C3×Dic3—21no—
97C2×C3×Dic3—41no—
98C3×Dic6—21no—
99C3×Dic6—41no—
100C3×Dic6—41no—
101C3×Dic6Q1612yes—
102C3⋊(C3⋊C8)—41no—
103C3⋊(C3⋊C8)—41no—
104C3×C3⋊C8—21no—
105C3×C3⋊C8—21no—
106C3×C3⋊C8—41no—
107#487—31no—
108#477—31no—
109#475—91no—
110C8×Q8—91no—
111#335—91no—
112#333—91no—
113#332—91no—
114C2×C3×Q8—22no—
115C2×C3×Q8—21no—
116C2×C3×Q8—21no—
117C2×C3×Q8—31no—
118C2×C3×Q8—61no—
119C2×Dic6—21no—
120C2×Dic6#6711yescharacteristic
121C2×C3×C8—31no—
122C2×C3×C8—31no—
123C2×C3×C8—61no—
124C3×C4⋊C4—21no—
125C3×C4⋊C4—21no—
126C3×C4⋊C4—21no—
127C3×C4⋊C4—21no—
128C3×C4⋊C4—31no—
129C3×C4⋊C4—41no—
130C3×C4⋊C4—41no—
131C3×C4⋊C4—61no—
132C3×C4⋊C4—61no—
133C3×C4⋊C4—61no—
134C3×C42—21no—
135C3×C42—61no—
136C3×C42C22×D311yescharacteristic
137C3×C42C22×D311yescharacteristic
138C4⋊Dic3—21no—
139C4⋊Dic3—21no—
140C4⋊Dic3#6711yescharacteristic
141Dic3⋊C4—21no—
142C4×Dic3—21no—
143C2×C3⋊C8—21no—
144C2×C3⋊C8—61no—
145C2×C3⋊C8—61no—
146C2×C3⋊C8—61no—
147C2×C3⋊C8—61no—
148C2×C3⋊C8—121no—
149C2×C3⋊C8—121no—
150C2×C3⋊C8#6711yescharacteristic
151C2×C3⋊C8#6711yescharacteristic
152C22×C32#12111yescharacteristic
153C32×C4—21no—
154C32×C4—81no—
155C32×C4C2×Q1611yescharacteristic
156C32×C4#13511yescharacteristic
157C32×C4#13511yescharacteristic
158C32×C4#13711yescharacteristic
159C3×Dic3—22no—
160C3×Dic3—41no—
161C3×Dic3—81no—
162C4⋊Q8—31no—
163C4⋊Q8D3211yescharacteristic
164C4×Q8—31no—
165#107—61no—
166#106—181no—
167C4⋊C8—31no—
168C4⋊C8—91no—
169C4⋊C8—91no—
170C4⋊C8—181no—
171#103—61no—
172#103—61no—
173#103—61no—
174#103—181no—
175C4×C8—31no—
176C4×C8—181no—
177C3×Q8—32no—
178C3×Q8—61no—
179C3×Q8—81no—
180C3×Q8—81no—
181C3×Q8—82no—
182C3×Q8—121no—
183C3×Q8—121no—
184C2×C3×C4—21no—
185C2×C3×C4—21no—
186C2×C3×C4—21no—
187C2×C3×C4—42no—
188C2×C3×C4—41no—
189C2×C3×C4—41no—
190C2×C3×C4—61no—
191C2×C3×C4—61no—
192C2×C3×C4—121no—
193C2×C3×C4D3×D411yescharacteristic
194C2×C3×C4D3×D411yescharacteristic
195C2×C3×C4#23711yescharacteristic
196C2×C3×C4#23711yescharacteristic
197C2×C3×C4C2×#6711yescharacteristic
198C2×C3×C4C2×#6711yescharacteristic
199C2×Dic3—21no—
200C2×Dic3—21no—
201C2×Dic3—41no—
202Dic6—21no—
203Dic6—41no—
204Dic6—41no—
205Dic6#21612yes—
206C3×C8—61no—
207C3×C8—61no—
208C3×C8—121no—
209C3⋊C8—21no—
210C3⋊C8—21no—
211C3⋊C8—41no—
212C3⋊C8—121no—
213C3⋊C8—121no—
214C3⋊C8—121no—
215C3⋊C8—121no—
216C3⋊C8—241no—
217C3⋊C8—241no—
218C2×C32#46411yescharacteristic
219C2×C32#46711yescharacteristic
220C2×C32#47111yescharacteristic
221C2×Q8—21no—
222C2×Q8—31no—
223C2×Q8—61no—
224C2×C8—31no—
225C2×C8—31no—
226C2×C8—61no—
227C2×C8—181no—
228C2×C8—181no—
229C4⋊C4—21no—
230C4⋊C4—21no—
231C4⋊C4—31no—
232C4⋊C4—61no—
233C4⋊C4—61no—
234C4⋊C4—61no—
235C42—61no—
236C42C2×D3211yescharacteristic
237C22×C3—21no—
238C22×C3#93811yescharacteristic
239C22×C3#93911yescharacteristic
240C3×C4—21no—
241C3×C4—21no—
242C3×C4—21no—
243C3×C4—21no—
244C3×C4—21no—
245C3×C4—21no—
246C3×C4—41no—
247C3×C4—62no—
248C3×C4—82no—
249C3×C4—81no—
250C3×C4—81no—
251C3×C4—121no—
252C3×C4—241no—
253C3×C4D3×Q1611yescharacteristic
254C3×C4#91811yescharacteristic
255C3×C4#91911yescharacteristic
256C3×C4#91911yescharacteristic
257C3×C4C2×#21611yescharacteristic
258C3×C4#95011yescharacteristic
259C3×C4#95011yescharacteristic
260C3×C4#95111yescharacteristic
261Dic3—22no—
262Dic3—41no—
263Dic3—81no—
264C32#168411yescharacteristic, complemented, Sylow
265Q8—32no—
266Q8—61no—
267Q8—81no—
268Q8—121no—
269Q8—121no—
270C2×C4—41no—
271C2×C4—61no—
272C2×C4—61no—
273C2×C4—121no—
274C2×C4#380511yescharacteristic
275C2×C4D3×#6711yescharacteristic
276C2×C4D3×#6711yescharacteristic
277C8—61no—
278C8—61no—
279C8—121no—
280C8—361no—
281C8—361no—
282C2×C3—21no—
283C2×C3—21no—
284C2×C3—21no—
285C2×C3#505111yescharacteristic
286C2×C3#505411yescharacteristic
287C2×C3#517711yescharacteristic
288C2×C3#518511yescharacteristic
289C2×C3#518811yescharacteristic
290C2×C3#520011yescharacteristic
291C22#6397011yescharacteristic
292C4—21no—
293C4—62no—
294C4—81no—
295C4—121no—
296C4—241no—
297C4D3×#21611yescharacteristic
298C4#6394311yescharacteristic
299C4#6394911yescharacteristic
300C4#6394911yescharacteristic
301C3—21no—
302C3#7115111yescharacteristic, complemented
303C3#7124111yescharacteristic, complemented
304C2#1059150611yescharacteristic
305C2#1059157811yescharacteristic
306C2#1059163511yescharacteristic
subgroupG/H
centerC22#63970
commutator, cosocleC32×C42C23
FrattiniC42C2×D32
FittingC32×C4⋊Q8C22
socle, prime radicalC22×C32#121
hypercenterC4⋊Q8D32
hypocenterC32#1684
LCSG ▷ C32×C42 ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ C4⋊Q8

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
264. C32#1684semidirect
302. C3#71151semidirect
303. C3#71241semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C3×#50932
C3×#50912
#105902852
C3⋊C8×Dic62
#105894122
#105890992
#105890922
#712413
#711513